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Sagot :
To determine which of the given equations corresponds to the quadratic formula provided, let's first dissect the components of the quadratic formula given:
[tex]\[ x = \frac{-8 \pm \sqrt{8^2-4(3)(-2)}}{2(3)} \][/tex]
The quadratic formula to solve [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, by comparing, we identify:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -8\)[/tex]
- The term inside the square root, [tex]\(8^2 - 4(3)(-2)\)[/tex], represents the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex], indicating:
[tex]\[ c = -2 \][/tex]
The quadratic equation [tex]\(3x^2 - 8x - 2 = 0\)[/tex] corresponds to the structure of the quadratic equation we started with and substituting [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = -2\)[/tex].
Now, let's match this with the provided options by transforming them to the form [tex]\(ax^2 + bx + c = 0\)[/tex]:
Option A: [tex]\(-2x^2 - 8 = 10x - 3\)[/tex]
Rearranging all terms to one side:
[tex]\[ -2x^2 - 10x - 8 + 3 = 0 \][/tex]
[tex]\[ -2x^2 - 10x - 5 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = -2\)[/tex], [tex]\(b = -10\)[/tex], [tex]\(c = -5\)[/tex]. This does not match our identified coefficients.
Option B: [tex]\(3x^2 - 8x - 10 = 4\)[/tex]
Rearranging:
[tex]\[ 3x^2 - 8x - 10 - 4 = 0 \][/tex]
[tex]\[ 3x^2 - 8x - 14 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], [tex]\(c = -14\)[/tex]. This does not match our identified coefficients.
Option C: [tex]\(3x^2 + 8x - 10 = -8\)[/tex]
Rearranging:
[tex]\[ 3x^2 + 8x - 10 + 8 = 0 \][/tex]
[tex]\[ 3x^2 + 8x - 2 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = 3\)[/tex], [tex]\(b = 8\)[/tex], [tex]\(c = -2\)[/tex]. This does not match our identified coefficients.
Option D: [tex]\(-2x^2 + 8x - 3 = 4\)[/tex]
Rearranging:
[tex]\[ -2x^2 + 8x - 3 - 4 = 0 \][/tex]
[tex]\[ -2x^2 + 8x - 7 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = -2\)[/tex], [tex]\(b = 8\)[/tex], [tex]\(c = -7\)[/tex]. This does not match our identified coefficients.
Following all the analysis, the equation that aligns with the quadratic formula provided is:
[tex]\(\boxed{B}\)[/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{8^2-4(3)(-2)}}{2(3)} \][/tex]
The quadratic formula to solve [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, by comparing, we identify:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -8\)[/tex]
- The term inside the square root, [tex]\(8^2 - 4(3)(-2)\)[/tex], represents the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex], indicating:
[tex]\[ c = -2 \][/tex]
The quadratic equation [tex]\(3x^2 - 8x - 2 = 0\)[/tex] corresponds to the structure of the quadratic equation we started with and substituting [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = -2\)[/tex].
Now, let's match this with the provided options by transforming them to the form [tex]\(ax^2 + bx + c = 0\)[/tex]:
Option A: [tex]\(-2x^2 - 8 = 10x - 3\)[/tex]
Rearranging all terms to one side:
[tex]\[ -2x^2 - 10x - 8 + 3 = 0 \][/tex]
[tex]\[ -2x^2 - 10x - 5 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = -2\)[/tex], [tex]\(b = -10\)[/tex], [tex]\(c = -5\)[/tex]. This does not match our identified coefficients.
Option B: [tex]\(3x^2 - 8x - 10 = 4\)[/tex]
Rearranging:
[tex]\[ 3x^2 - 8x - 10 - 4 = 0 \][/tex]
[tex]\[ 3x^2 - 8x - 14 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], [tex]\(c = -14\)[/tex]. This does not match our identified coefficients.
Option C: [tex]\(3x^2 + 8x - 10 = -8\)[/tex]
Rearranging:
[tex]\[ 3x^2 + 8x - 10 + 8 = 0 \][/tex]
[tex]\[ 3x^2 + 8x - 2 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = 3\)[/tex], [tex]\(b = 8\)[/tex], [tex]\(c = -2\)[/tex]. This does not match our identified coefficients.
Option D: [tex]\(-2x^2 + 8x - 3 = 4\)[/tex]
Rearranging:
[tex]\[ -2x^2 + 8x - 3 - 4 = 0 \][/tex]
[tex]\[ -2x^2 + 8x - 7 = 0 \][/tex]
Comparing coefficients:
[tex]\(a = -2\)[/tex], [tex]\(b = 8\)[/tex], [tex]\(c = -7\)[/tex]. This does not match our identified coefficients.
Following all the analysis, the equation that aligns with the quadratic formula provided is:
[tex]\(\boxed{B}\)[/tex]
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